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Re: Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

When you define the Atiyah algebroid, are you taking the anchor map to be to TM, or to M (i.e. composing ρ and p)?

Have you looked at the fact that the splitting

TP/G*→P×GLie(G)=adP

of

adP→TP/G*

in

0→adP→TP/G*→TM→0

also gives a connection (actually the connection form)? The s.e.s. is from “Complex analytic connections in fibre bundles” and is where your Atiyah algebroid comes from (I know you know this).

The nice thing about using this Atiyah definition of connection is it gives a “closer” link to the usual horizontal subspace/connection form defintion of connection. We just take a left or right splitting of the above s.e.s. to get the connection.

All this should lift nicely to 2-bundles, once a decent definition of 2-vector bundle is figured out (I think Danny may be working on this).

Re: Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

Yes, as the anchor is always to TM. By displaying the projection arrow TM→M I didn’t mean anything deep.

Have you looked at the fact that […]

No, not really. But I do believe that this should be important for the following reason:

We are dealing with these two equivalent aspects of ordinary connections:

a) something that allows to lift, namely vectors/paths in the base space M to vectors/paths in the total space P

b) something that defines horizontal subspaces of TP.

Here a) is really defining a connection in terms of its holonomy. Categorifying this leads to the constraint of vanishing fake curvature.

Possibly the non-fake flat situation as obtained for nonabelian bundle gerbes with connection and curving can be understood as a categorification of b) instead of a). I believe Danny was hinting at that (but I might be wrong).

In the case of ordinary bundles b) does allow to lift paths to total space, too.

In the case of categorified bundles (gerbes) it is no longer obvious that the categorification of b) allows to lift surfaces from base space to total space.

I expect that adding to the categorification of b) the requirement that it should be possible to ‘lift surfaces’ will be precisely equivalent to the categorification of a) and hence of in addition requiring fake flatness.

once a decent definition of 2-vector bundle

As was mentioned here recently, as far as tangent 2-bundles are concerned it should be clear what one has to do:

Let S be a smooth category (smooth space of objects, smooth space of morphims and all maps between them smooth maps). S is a ‘2-space’. Its tangent 2-bundle should be the category TS with

(1)Obj(TS)=T(Obj(S))

(2)Mor(TS)=T(Mor(S))

(3)sTS=dsS

(4)tTS=dtS

(5)∘TS=d∘S.

(Here, for instance, sTS denotes the source map in TS and dsS the differential of the source map in S.)

This is a category internalized in the category of vector bundles and hence qualifies as a good notion of 2-vector bundle. And it exists for every smooth category S. This should be all that is needed for the above purposes.

Re: Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

Could you expand on that? I am afraid I don’t know what you have in mind. Are you talking of Grassmannians in the sense of spaces of k-dimensional subspaces of vector spaces or something else? Sorry.

if we let “dim S” →∞

This might actually be the ‘generic’ case, in a sense. Many intersting smooth spaces of morphism will be infinite-dimensional. For instance if morphisms are paths in the space of objects.

a nice canonical example of a 2-bundle

I need to think more about this. One issue seems to be that such a tangent 2-bundle is not at all of the form of those 2-bundles which are related to gerbes, unless I am confused. I.e. it does not locally look like S×F for F the typical fiber category. Or does it somehow?

—-
One issue seems to be that such a tangent 2-bundle is not at all of the form of those 2-bundles which are related to gerbes, unless I am confused. I.e. it does not locally look like S×F for F the typical fiber category.
—-

That’s good - we need non-gerbe examples of 2-bundles. From the algebraic geometric side of things (a la Grothendieck, Deligne-Mumford etc) general stacks seem to me to be less “nice” than gerbes. Looking at the axioms for a gerbe:

The first of these means that there is a cover such that the fibre (“set of sections”) over each element of the cover is non-empty (something like local triviality), and the second means that all the automorphism groups of objects in the fibres are isomorphic (i.e. we have a well defined fibre)

So in a sense you are correct about the nasty properties of these non-gerbe 2-bundles, but hey, I’m sure someone will find a way around this. Just remember what all these objects were invented for: gerbes for degree 2 nonabelian cohomology, 2-bundles for categorifying bundles. Both have done their job. It’s just nice they overlap a bit.

Re: Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

I might add that the different notions of ‘2-bundle’ playing a role here seem to be due to two different branches of categorification.

Roughly, whenever there is a concept A one can

1) either study A internalized (strictly) in Cat

2) or study categories internalized in the category of A .

There is an amazing theorem that 1) and 2) agree ifA is what is called essentially algebraic. But in general they need not agree.

The principal 2-bundles as defined originally by Toby Bartels are like principal bundles internalized in Cat, i.e. a categorification of a bundle according to procedure 1).

The tangent 2-bundles that we were talking about however are categories internalized in the category of vector bundles and hence are (vector) bundles categorified according to procedure 2).

Categorification is a little bit like quantization in that it may give different and inequivalent results when applied to different but equivalent items.

So what would a 2-section of a 2-bundle in the sense 2) be? That seems to be the central question, because the relation to stacks should be that a 2-bundle has a stack of 2-sections (instead of just a sheaf of 1-sections).

I haven’t really thought hard enough, but a straightforward guess is that a 2-section of the tangent 2-bundle TS that we talked about must be just an ordinary section f of Mor(TS).

That, however, makes it seem unlikely that these 2-sections form a stack, somehow.

Maybe one should add the (nontrivial) requirement that the image of f under source and target maps s and t are sections of Obj(TS).

Re: Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

I haven’t really thought hard enough, but a straightforward guess is that a 2-section of the tangent 2-bundle TS that we talked about must be just an ordinary section f of Mor(TS).

That, however, makes it seem unlikely that these 2-sections form a stack, somehow.

Maybe one should add the (nontrivial) requirement that the image of f under source and target maps s and t are sections of Obj(TS).

I would have thought that a 2-section of a 2-map F:E→M would be a functor S:M→E such that F∘S⇒idM. I started working on something like this but got distracted ;) This was going to be my starting point of “2-sections of a 2-bundle form a stack”.

Re: Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

Hi David,

sorry for my slow responses. I am currently distracted by a couple of things, like preparing for my disputation (which I should be doing right now instead of chatting) and finding a place to live in Hamburg (a task that has now successfully and satisfactorily been finished).

I would have thought that a 2-section of a 2-map F:E→M would be a functor S:M→E such that F∘S⇒idM.

True. In my last comment I was missing the fact that such a projection map F is available here.

So at the rough level at which I have thought about this stuff, it seems that given any category E internalized in the category of vector bundles, we get an internal projection functor E→pB from it to a category B internalized the category of smooth spaces as follows:

Let

(1)EObj→pObjBObj

be the vector bundle of objects of E and let

(2)EMor→pMorBMor

be the vector bundle of morphisms of E.

The source, target and composition maps in E are vector bundle morphisms which ristrict to smooth maps on the base spaces BObj and BMor. Calling these restrictions sB, tB and ∘B, respectively, we seem to get a ‘base category’ B of E with

(3)B={BObj,BMor,sB,tB,∘B}

in the (hopefully) obvious way. By simply forgetting about the fibers in the bundles involved in E we should get a projection functor

(4)E→pB

and hence arrive at a 2-bundle in the sense 1) of my previous comment.

Ok, I didn’t realize this simple fact last time. Now with the p-functor available, it is clear what a 2-section should be, namely, as you say, a 2-functor

(5)B→sE

such that

(6)B→sE→pB⇔λB→IdB.

But now, isn’t it true that such a 2-section s must come (at least in the case where the above isomorphism λ is the identity) from an ordinary section of
EMor→pMorBMor ?

I guess so. But actually the example of principal 2-bundles shows that the interesting aspect of 2-sections is in their morphisms. Is there an interesting structure on the natural transformations

(7)B→s1E⇒κB→s2E

?

It is noteworthy that the base category B is an honest category with, in general, nontrivial morphisms. This means that the ‘2-sheaf’ of 2-sections s is not in an obvious way a stack. That’s because a stack is what I would call a pseudo-functor instead of a true 2-functor. Meaning that it is something from a 1-category (of ‘open sets’) to the 2-category Cat (or Groupoids if you like) instead of something whose domain is a 2-category itself.

Here it seems we need a 2-category of 2-‘open sets’ of B, namely something like the 2-category of ‘open subcategories’ of B.

I once talked more about that somewhere on the web, but haven’t really made much progress with it. But it is a general qestion that I would someday like to know the answer to:

Why stacks instead of ‘2-sheafs’?

Not that I would find stack theory so trivial that I would be yearning for something more intricate. But from the point of view of categorification a stack is a an inconsequent generalization of a sheaf, it seems.

Re: Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

Probably. But it bites us here. On the other hand, in special cases we can still hope to get a stack of 2-sections.

Namely if the base category B has the following property.

To each open set Ui⊂BObj let

(1)B∣Ui:=(s−1(Ui))∩(t−1(Ui))

be the subcategory of B consisting of all morphism starting and ending in Ui. If all of B is generated from

(2)⋃iB∣Ui

for some covering {Ui}i of BObj then it does make good sense to define the category of 2-sections on Ui to be that of 2-sections B∣Ui→sE. Hence we do get a fibered category of 2-sections. With a little luck this is a stack.